Applied Scientific Research

, Volume 17, Issue 6, pp 422–438 | Cite as

Determination of the orthotropic plate parameters of stiffened plates and grillages in free vibration

  • K. T. Sundara Raja Iyengar
  • R. Narayana Iyengar
Article

Summary

It is well known that the analysis of vibration of orthogonally stiffened rectangular plates and grillages may be simplified by replacing the actual structure by an orthotropic plate. This needs a suitable determination of the four elastic rigidity constants Dx, Dy, Dxy, D1 and the mass \(\bar \rho \) of the orthotropic plate. A method is developed here for determining these parameters in terms of the sectional properties of the original plate-stiffener combination or the system of interconnected beams. Results of experimental work conducted on aluminium plates agree well with the results of the theory developed here.

Notation

a, b

span and width of stiffened plates, grillages and orthotropic plates

c, \(\bar c\)

spacing of beams in the y and x directions

D

rigidity of an unstiffened isotropic plate

Dx, Dy

rigidities in the x and y directions of an orthotropic plate

Dxy

torsional rigidity of an orthotropic plate

D1

a parameter associated with a poisson type ratio

e

depth of the common neutral axis of the beam-plate combination below the middle surface of the plate

E, G

modulus of elasticity and modulus of shear

H

=D1+2Dxy

i, j

integers

I, Ī, Ii, Ij

moment of inertia of beams

J, Ji, Jj

polar moment of inertia of beams

m, n

integers referring to mode numbers in the x and y directions

pm n

natural frequency of an orthotropic plate

r, s

number of beams in the transverse and longitudinal direction

Tg, To, Ts

kinetic energy of grillages, orthotropic plates and stiffened plates

u

frequency parameter of grillages

Vg, Vo, Vs

potential energy of grillages, orthotropic plates and stiffened plates

W=W(x, y, t)

deflection of an orthotropic plate

w=w(x, y)

amplitude of vibration of orthotropic plates, stiffened plates and grillages

x, y, xi, yi

cartesian coordinates

Xm, Yn

beam eigen functions

Xmj, Yni

Xmand Ynat x=xjand y=yi

X′m, Y′n; Xm, Yn

first and second derivatives of Xmand Ynrespectively

Ymn

plate eigen functions

αn, βn

parameters occurring in the expression for the beam functions

γ, \(\bar \gamma \), γi, γj

mass per unit length of beams

λmn

=\(\omega _{m n} b^2 \sqrt {{\rho \mathord{\left/ {\vphantom {\rho D}} \right. \kern-\nulldelimiterspace} D}} \) frequency parameter of beam and slab bridges

μ

=\({H \mathord{\left/ {\vphantom {H {\sqrt {D_x D_y } }}} \right. \kern-\nulldelimiterspace} {\sqrt {D_x D_y } }}\)

ν

Poisson's ratio

ρ, \(\bar \rho \)

mass per unit area of unstiffened plates and orthotropic plates

ωmn

natural frequency of stiffened plates and grillages

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Copyright information

© Martinus Nijhoff 1967

Authors and Affiliations

  • K. T. Sundara Raja Iyengar
    • 1
  • R. Narayana Iyengar
    • 1
  1. 1.Department of Civil EngineeringIndian Institute of ScienceBangalore-12India

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